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Chapter 34: Problem 56

The wavelengths of a spectral line series tend to a limit as \(n_{1} \rightarrow \infty .\) Evaluate the series limit for (a) the Lyman series and (b) the Balmer series in hydrogen.

Short Answer

Expert verified
The series limit for the Lyman series in hydrogen is approximately 91.18 nm, and for the Balmer series, it is approximately 364.6 nm.

Step by step solution

01

Identify Given Information

From the problem, we know that we're dealing with a hydrogen atom and we're also given that for the series limit, \( n_1 \rightarrow \infty \). The lower limit for both the Lyman and Balmer series is 1 and 2 respectively. The Rydberg constant for hydrogen \( (R_H) \) is approximately \( 1.097 × 10^7 \, m^{-1} \).
02

Apply the Rydberg Formula for Lyman series

The Lyman series corresponds to transitions where \( n_2 = 1 \). We want to find the limit as \( n_1 \rightarrow \infty \). So, apply the Rydberg formula and take the limit as \( n_1 \rightarrow \infty \), which simplifies to \( \lim_{{n_1}\to\infty} \left (\frac{1}{\lambda} = R_H \left ( \frac{1}{1^2} - \frac{1}{n_1^2} \right ) \right ) \), which gives \( \frac{1}{\lambda} = R_H \). Thus, \( \lambda = \frac{1}{R_H} \).
03

Calculate the Series Limit for Lyman Series

Now we substitute the given value of \( R_H \) into the formula obtained in step 2 to get the series limit for the Lyman series: \( \lambda = \frac{1}{R_H} = \frac{1}{1.097 × 10^7 \, m^{-1}} \approx 91.18 \, nm \).
04

Apply the Rydberg Formula for Balmer series

The Balmer series corresponds to transitions where \( n_2 = 2 \). We want to find the limit as \( n_1 \rightarrow \infty \). So, apply the Rydberg formula and take the limit as \( n_1 \rightarrow \infty \), which simplifies to \( \lim_{{n_1}\to\infty} \left (\frac{1}{\lambda} = R_H \left ( \frac{1}{2^2} - \frac{1}{n_1^2} \right ) \right ) \), which gives \( \frac{1}{\lambda} = \frac{R_H}{4} \). Thus, \( \lambda = \frac{4}{R_H} \).
05

Calculate the Series Limit for Balmer Series

As in step 3, substitute the given value of \( R_H \) into the formula obtained in step 4 to get the series limit for the Balmer series: \( \lambda = \frac{4}{R_H} = \frac{4}{1.097 × 10^7 \, m^{-1}} \approx 364.6 \, nm \).

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